Nonlinear Separation Functions Research Articles

Nonlinear, parametrically excited dynamics of two-stage spur gear trains with mesh stiffness fluctuation

This work studies the nonlinear, parametrically excited dynamics of two spur gear pairs in a two-stage counter-shaft configuration. The dynamic model includes parametric excitations and contact loss of both tooth meshes with two different mesh frequencies. The time-varying mesh stiffnesses and nonlinear tooth separation functions are reformulated into forms suitable for perturbation analysis. The periodic steady-state solutions are obtained by the method of multiple scales and compared against a semi-analytical harmonic balance method as well as numerical integration for fundamental and subharmonic resonances for ranges of system parameters. The interaction of the two meshes is found to depend strongly on the relation of the two mesh periods. The dynamic influences of design parameters, such as shaft stiffness, mesh stiffness variations, contact ratios, and mesh phasing, are discussed. The closed-form solutions provide design guidelines in terms of the system parameters.

Nonlinear Separation Approach to Constrained Extremum Problems

In this paper, by virtue of a nonlinear scalarization function, two nonlinear weak separation functions, a nonlinear regular weak separation function, and a nonlinear strong separation function are first introduced, respectively. Then, by the image space analysis, a global saddle-point condition for a nonlinear function is investigated. It is shown that the existence of a saddle point is equivalent to a nonlinear separation of two suitable subsets of the image space. Finally, some necessary and sufficient optimality conditions are obtained for constrained extremum problems.

Periodic Activation Function and a Modified Learning Algorithm for the Multivalued Neuron

In this paper, we consider a new periodic activation function for the multivalued neuron (MVN). The MVN is a neuron with complex-valued weights and inputs/output, which are located on the unit circle. Although the MVN outperforms many other neurons and MVN-based neural networks have shown their high potential, the MVN still has a limited capability of learning highly nonlinear functions. A periodic activation function, which is introduced in this paper, makes it possible to learn nonlinearly separable problems and non-threshold multiple-valued functions using a single multivalued neuron. We call this neuron a multivalued neuron with a periodic activation function (MVN-P). The MVN-Ps functionality is much higher than that of the regular MVN. The MVN-P is more efficient in solving various classification problems. A learning algorithm based on the error-correction rule for the MVN-P is also presented. It is shown that a single MVN-P can easily learn and solve those benchmark classification problems that were considered unsolvable using a single neuron. It is also shown that a universal binary neuron, which can learn nonlinearly separable Boolean functions, and a regular MVN are particular cases of the MVN-P.

Solving the XOR and parity N problems using a single universal binary neuron

A universal binary neuron (UBN) operates with complex-valued weights and a complex-valued activation function, which is the function of the argument of the weighted sum. The activation function of the UBN separates a whole complex plane onto equal sectors, where the activation function is equal to either 1 or −1 depending on the sector parity (even or odd, respectively). Thus, the UBN output is determined by the argument of the weighted sum. This makes it possible the implementation of the nonlinearly separable (non-threshold) Boolean functions on a single neuron. Hence, the functionality of UBN is incompatibly higher than the functionality of the traditional perceptron. In this paper, we will consider a new modified learning algorithm for the UBN. We will show that classical nonlinearly separable problems XOR and Parity n can be easily solved using a single UBN, without any network. Finally, it will be considered how some other important nonlinearly separable problems may be solved using a single UBN.

Minimal realization of arbitrary switching functions with 2-level networks of isodistinct and isobaric threshold gates

Abstract This paper deals with the synthesis of 2-level networks of threshold logic elements for realizing non-linearly separable switching functions. Gate minimality has been ensured for the types of gates used. The method is straightforward and can be programmed in a computer.

Minimum Two-Level Threslold Gate Realizations

This paper deals with the synthesis of two-level networks of threshold gate logic elements for realization of nonlinearly separable switching functions. The realization obtained contains the minimum number of threshold logic elements possible for a two-level realization. An algorithm based on the tree procedure of Coates and Lewis is developed which can be used to obtain the desired network realization for a given switching function. The function may be incompletely specified.

Test and realization of 2-separable switching functions†

In this paper a method of test and realization of non-linearly separable switching functions that are 2-rcalizable, i.e. realizable by a network of 2-threshold logic elements is presented. The positive linear dependences among the rows of the coefficient matrix of the set of inequalities representing the function are first determined and systematically removed using a distribution table. All possible solutions are readily obtained. A weight matrix representation of a network of threshold logic elements is used. An example is worked out.

Geometrical and Statistical Properties of Systems of Linear Inequalities with Applications in Pattern Recognition

This paper develops the separating capacities of families of nonlinear decision surfaces by a direct application of a theorem in classical combinatorial geometry. It is shown that a family of surfaces having d degrees of freedom has a natural separating capacity of 2d pattern vectors, thus extending and unifying results of Winder and others on the pattern-separating capacity of hyperplanes. Applying these ideas to the vertices of a binary n-cube yields bounds on the number of spherically, quadratically, and, in general, nonlinearly separable Boolean functions of n variables. It is shown that the set of all surfaces which separate a dichotomy of an infinite, random, separable set of pattern vectors can be characterized, on the average, by a subset of only 2d extreme pattern vectors. In addition, the problem of generalizing the classifications on a labeled set of pattern points to the classification of a new point is defined, and it is found that the probability of ambiguous generalization is large unless the number of training patterns exceeds the capacity of the set of separating surfaces.